Elastic interfacial instability

Chapter 4. Viscous and elastic instabilities in confined geometries

(a) The meniscus instability of a thin elastic film in a peel geometry. b = 150µm, G = 1×10⁶Pa. The flexural rigidity of the cover glass increases from top (0.02 Nm) to bottom (0.2 Nm). From reference [47].

(b) 2D ripple pattern observed upon contact when approaching a thin elastomeric layer (b= 12.1µm) to a parallel glass slide. From reference [79].

Figure 4.7.: Contact instabilities in confined elastic systems.

came to rest. Figure4.7(a)shows the wavelength and amplitude of the undulations for fixed thickness b and shear modulus G but for different flexural rigidities D of the cover slip.

The main result of this study was that the wavelength of the instability increases linearly with the film thickness, but stays independent both of the flexural rigidity of the cover slip and the shear modulus of the elastic layer. This elastic instability involves no material flow.

A similar situation was investigated by M¨onch and Herminghaus in 2001 [79].

They studied the pattern formation when a very thin layer of a soft polymer is fixed on a lower rigid glass substrate and approached to an upper rigid glass substrate.

The elastomeric film and the contactor were parallel in this study. Film thicknesses ranged from 7µm to 60µm. From a critical distance on, the authors observed a 2D ripple pattern with a typical wavelengthλ. As in the study by Ghatak et al.,λ depended solely on the film thickness.

A mathematical model for the plane strain case has been developed by M¨onch et al. [79] as well as by Shenoy and Sharma in 2001 [105]. We give here the basic principles and main results as reported in [105]. An elastomeric film of thickness b is strongly bonded to a rigid substrate. A flat rigid contactor is placed in the distance dfrom the free film surface, and the free surface of the film is allowed to deform. The interaction forces between the two surfaces, which tend to destabilize the stystem, are in competition with the elastic restoring forces in the film. The authors draw the energy balance upon approach of the contactor,

E_total =E_elastic+Einteraction+E_surface. (4.41) The interaction energy is not specified in their model and can originate for exam-

4.3. Elastic interfacial instability ple from van der Waals interactions between the free surface and the contactor.

The authors show that a homogeneous solution for the displacement field u in the elastomeric film exists, so that the stresses in the film are equal everywhere. This equilibrium stress field minimizes the total system energy equation (4.41) in a lin- earized form. Subsequently, they test the stability of this homogeneous solution to disturbing displacement fields of the form u_j = e^ikx1u_j(x₂). They find that for sufficiently small distancesd, the homogeneous solution is unstable and undulations of the contact line appear. The critical wavenumber k_c is determined by a rather complex function of the thicknessb, the Poisson’s ratioν, the surface tension γ and the film’s shear modulusG. For the relevant experimental conditions for elastomers ν→0.5 andγ/Gh¿1 though, this function reduces to the simpler form

b k_c= 2.12−2.86(1−2ν)−2.42 γ

Gh . (4.42)

It is obvious that the wavelengthλ_c= 2π/k_cdoes solely depend on the film thickness for common elastomeric films with the typical valuesν ≈0.5, G≈1 MPa, b >1µm, andγ ≈0.1^J/m².²

The main results of the model are:

• The destabilizing wavelength λ is independent of all system parameters but the film thickness.

• The wavelength does not depend on the exact nature of the interaction. The destabilization is triggered by the interaction energy but the mode selection is not influenced.

• The surface tension has a stabilizing influence, but is negligible in common experimental situations.

• If one includes viscous stresses into the stability analysis, the critical viscoelas- tic mode recovers the critical elastic mode for the typical experimental param- eters.

A broader study involving different setups such as circular and rotating geometries has been published in 2003 [46]. As mentioned before, there exists a critical film thickness b_c below which the contact line in the peeling geometry is instable and starts to undulate. Performing a systematic study with different rigidities of the cover plate and shear moduli of the elastic film, Ghatak and Chaudhury found experimentally that

b_c∝ µD

G

¶_1/3

, (4.43)

2Very recently, Arun et al. investigated the contact instability upon approach of an upper glass plate to a viscoelastic material in an electrical field [7]. In this study, the parameters are chosen in such way that the additional terms in equation4.42become important compared tobkc= 2.12.

A transition from viscous to elastic behavior is reflected in a change of the wavelength of this surface instability.

Chapter 4. Viscous and elastic instabilities in confined geometries

Figure 4.8.: In the peeling geometry, the critical thickness depends on the bending stiffness of the cover plateD and on the shear modulus of the filmG, taken from [46].

as displayed in figure4.8. This dependence has been explained theoretically in 2005 by Ghatak et al. [48] and in 2006 by Adda-Bedia and Mahadevan [1]. We follow here the line of argumentation of the latter reference, in which the peel geometry is investigated and not, as before, the plain strain geometry.

We consider again a peeling geometry where a plate with rigidityD is in contact with an elastomer with shear modulusG. For a relatively thick film, the peeling front is a straight line and does not show undulations. The cover plate shall be deformed by the small length δ over a lengthscale l_p. The deformations decay exponentially into the bulk of the elastomer, also with the typical lengthscalel_p. The penetration depth can be determined studying the energy balance in this situation. When the contact line is not moving, the bending energy per unit area in the plate is balanced by the bending energy per unit area in the elastomer,

Dδ² l³_p =G

µδ l_p

¶₂

l²_p. (4.44)

From equation 4.44it follows for the penetration depth that l_p∝

µD G

¶_1/3

. (4.45)

If the film thickness b is decreased, at some point the deformations cannot decay overl_p anymore but are forced to decay over b < l_p. This introduces a confinement parameterα:

α= µ D

Gb³

¶_1/3

. (4.46)

For α¿ 1, the system is unconfined, the film can relax deformations over l_p along its thickness and a straight peeling front is possible. On the other hand, for αÀ1, the penetration depth isbinstead ofl_p. This creates large stresses around the crack

4.3. Elastic interfacial instability and the contact line explores other configurations to relieve the stored elastic energy.

It turns out that shear formations are favored over normal (bulk) deformations. The energy cost for the creation of surface is compensated by the elastic energy released in the film, and the contact line prefers to undulate [1]. Performing a linear stability analysis, the authors were able to determine a critical wavenumberk_cb≈1.85 and a critical confinementα_c≈21 above which the crack front is unstable. These values are in good agreement with experimental results from reference [46], α_c ≈ 18 and k_cb≈1.57.

The instabilities described above are observed in astatic peeling geometry. How- ever, the observation of fingering instabilities in a dynamic debonding situation has also been reported, for example in the situation of an adhesive layer debonded from a silicone substrate [57], a layer of latex (PEHA) debonded from steel [19], or in dynamic peeling [115].

Part II.

Materials and methods